What'd be the best way to make a rim?
I think this sort of thing is much easier to work in the flat, keeping the vertex count low, and working under a Subdivision modifier

So, I've made a nice pot, and a lid. But the lid doesn't have an inside, and needs to fit a recess in the pot.


In Edit Mode CtrlD steal (duplicate) a profile loop, lying in the XZ plane, from the original pot, and P split it off into a new object.
Do the same with the lid.
Join the new profiles CtrlJ into a new object.
Assign a Screw and Subdivision modifier to the new combined profile, to make it back into a pot-with-a-lid.
You can now edit the profiles in place -  if you don't show the modifiers in Edit Mode, you can flip back and forth Tab to Object mode to see how your work is going.


The picture shows the original profile in the vicinity of the rim, and 2 steps in the edit - using E extrusion on vertices to draw out new edges. Still in Edit, to make the inner dome,I created a new Circle mesh (unfilled) to add to the profile, moved and scaled it into the right place, deleted most of it, and O (Connected) proportionally edited it to keep it smooth inside. (Always keeping everything flat in XZ)




You have the choice of if or when to apply the modifiers, add bevels, whatever you like.




In short, this recommendation is: edit the profile, not the pot.